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  • Parity-constrained Triangulations with Steiner points

    Victor Alvarez∗

    October 31, 2012

    Abstract

    Let P ⊂ R2 be a set of n points, of which k lie in the interior of the convex hull CH(P) of P . Let us call a triangulation T of P even (odd) if and only if all its vertices have even (odd) degree, and pseudo-even (pseudo-odd) if at least the k interior vertices have even (odd) degree. On the one hand, triangulations having all its interior vertices of even degree have one nice property; their vertices can be 3-colored, see [1, 2, 3]. On the other hand, odd triangulations have recently found an application in the colored version of the classic “Happy Ending Problem” of Erdős and Szekeres, see [4].

    In this paper we show that there are sets of points that admit neither pseudo- even nor pseudo-odd triangulations. Nevertheless, we show how to construct a set of Steiner points S = S(P ) of size at most k

    3 +c, where c is a positive constant, such that

    a pseudo-even (pseudo-odd) triangulation can be constructed on P ∪ S. Moreover, we also show that even (odd) triangulations can always be constructed using at most n

    3 + c Steiner points, where again c is a positive constant. Our constructions have

    the property that every Steiner point lies in the interior of CH(P ).

    1 Introduction

    Let P ⊂ R2 be a set of n points in general position and let Γ : P → {0, 1} be an assignment of parities to the elements of P , where 0 means even and 1 means odd. Let G be a straight-edge plane graph with vertex set P . We say that a vertex v ∈ P of G is happy, with respect to G, if and only if the degree of v in G is of the parity assigned to v by Γ. If a vertex is not happy w.r.t. G we will say that v is unhappy. If the graph G is clear from context we will just say that vertices are happy or unhappy without referring to G.

    Given P and Γ, the problem of finding plane graphs on P that maximize the number of happy vertices has recently received some attention. In [5] it was shown that one can always construct a tree, a 2-connected outerplanar graph, and a pointed pseudo- triangulation that makes all but at most three vertices happy. For the class of triangu- lations it was shown that one can always construct one that makes essentially 2n

    3 of its

    ∗Fachrichtung Informatik, Universität des Saarlandes, alvarez@cs.uni-saarland.de.

    1

  • vertices happy, but a configuration of points with parities was also shown where at least n 108

    vertices remain unhappy regardless of the chosen triangulation. The construction of this lower bound requires the use of both parities, but the authors pointed out that there are two particular cases that might accept triangulations with as many as n−o(n) happy vertices. These two particular cases are the ones where the parities assigned to the elements of P by Γ are either all even or all odd respectively. However, showing that in those particular cases n − o(n) vertices can be made happy turned out to be very challenging. In the same paper the authors showed that in the all-even case, a tri- angulation that makes at least 2n

    3 vertices happy can always be constructed. They also

    showed that in the all-odd case a 10 13

    fraction of happy vertices can always be ensured.

    These two special cases, all even or all odd, are of significant interest since they have interesting properties and/or applications. For example, it is well known that a connected graph G having all its vertices of even degree is Eulerian. If on top of it G happens to be a triangulation as well, then G is also 3-colorable, see [1, 2] for a general reference on 3-colorable planar graphs. Those two properties are usually considered “nice” in a graph, and they are characterized only by the parity of the degree of its vertices. For 3-colorability of triangulations a slightly weaker characterization is known: T is a triangulation having all its interior vertices of even degree if and only if T is 3-colorable, see [3] for example. The application related to the all-odd case is a little bit more intricate and we refer the reader to [4] where this application is shown.

    Let P be as before. We will say that a triangulation T of P is even, or odd, if and only if the degree of every vertex of T is even, or odd respectively. If only the interior vertices of T are even, or odd, we will say that T is pseudo-even, or pseudo-odd respectively. This defines four kinds of triangulations of P : Even, pseudo-even, odd, pseudo-odd.

    In this paper we attack the following problem: Given P and one kind T of triangula- tions of the four mentioned above, construct a set S = S(P,T ) such that a triangulation of kind T can always be constructed on P ∪ S.

    Thus, the problem attacked in this paper can be seen as the Steiner-point version of the ones regarding triangulations presented in [5]. With this idea in mind, the following lemma is worth noting:

    Lemma 1. There are arbitrarily large sets of points that, without the use of Steiner

    points, admit neither pseudo-even nor pseudo-odd triangulations.

    Proof. Let P be a set of points like the one shown to the left in Figure 1 where the size of the convex polygon Q shown in gray is ≡ 1 mod 3. Let v ∈ P be the only point not in Q. It is clear that in any triangulation of P , point v must be adjacent to every vertex of Q, that is, without a triangulation of Q, every vertex of Q has degree three.

    Now, it is well known that every triangulation of a polygon has at least two “ears”, i.e., a triangle formed by three consecutive vertices of the polygon. This means that, regardless of what triangulation of Q we choose, there will always be a vertex of Q whose adjacencies are only its two neighbors in Q and v. Thus no pseudo-even triangulation of P exists. See to the right in Figure 1.

    2

  • v

    Q

    v

    u w u w

    Figure 1: To the left we have a configuration in which all shown adjacencies are forced, and it accepts neither pseudo-even nor pseudo-odd triangulations. To the right we show in red one of the ears of the shown triangulation of Q.

    To show that P does not admit a pseudo-odd triangulation either it suffices to show that regardless of what triangulation of Q we choose, there will always be at least one interior point of P having even degree. It is not hard to verify that in an even triangulation, the size of the outer face must be ≡ 0 mod 3, a proof can be found in [3]. Since |Q| ≡ 1 mod 3, then Q does not admit an even triangulation, so in every triangulation of Q there will be at least two vertices of odd degree, there must be an even number of them. So assume that there is a triangulation of Q in which the only two vertices of odd degree are the two neighbors u,w of v in CH(P ). Thus we could add a point p outside CH(P ), below the edge uw, and add the adjacencies pu, pw. This implies that the set of points Q′ = Q ∪ {p} has an even triangulation, where upw is an ear. But |Q′| ≡ 2 mod 3, which clearly contradicts the necessary condition on the size of the outer face of an even triangulation. Therefore, in any triangulation of Q there must be at least one interior point q ∈ P of odd degree. The force adjacency qv implies that q gets even degree in a triangulation of P , which is what we wanted to prove. �

    2 Our contribution

    By Lemma 1 the use of Steiner points is sometimes necessary if we insists in constructing any of the four kinds of triangulations mentioned before (even, pseudo-even, odd, pseudo- odd). The relevant issue now is not whether we can construct the triangulations we are interested in, but rather with how many Steiner points can we achieve such constructions, the less, the better. The results we are going to show are the following:

    Theorem 1. Let P ⊂ R2 be a set of n points in general position where k of them are interior points. Then a set S = S(P ) of interior Steiner points of size at most

    k+2 3

    +2 can always be constructed such that P ∪ S accepts a pseudo-even triangulation.

    Theorem 2. Let P be as before. Then a set S = S(P ) of interior Steiner points of size at most

    n+1 3

    +1 can always be constructed such that P∪S admits an even triangulation.

    3

  • Theorem 3. Let P ⊂ R2 be a set of n points in general position where k of them are interior points. Then a set S = S(P ) of interior Steiner points of size at most

    k 3

    + c, with c a positive constant, can always be constructed such that P ∪S accepts a pseudo-odd triangulation.

    Theorem 4. Let P be as before. Then a set S = S(P ) of interior Steiner points of size at most

    n−1 3

    + c, with c a positive constant, can always be constructed such that P ∪S admits an odd triangulation.

    The proofs of all theorems will be constructive. The rest of the paper is divided as follows: In Section 3 we start with some preliminaries. In Section 4 and 5 we show algorithms that imply Theorems 1, 2, and Theorems 3, 4 respectively. We close the paper in Section 6 we some observations and conclusions.

    3 Pre-processing of P

    Let us quickly recall that given a polygon P, a vertex v of P is called reflex if the internal angle at v is larger than π. We will call it convex otherwise. Also, by a suitable rotation of P we can assume that the vertex v of CH(P ) with the lowest y-coordinate is unique.

    The following pre-processing of P will be done: Using v as a pivot we will sort each interior point of P by its angle with respect to v. Let p1, . . . , pk, be a labeling, from left

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